# American Institute of Mathematical Sciences

March  2016, 36(3): 1603-1628. doi: 10.3934/dcds.2016.36.1603

## Infinitely many solutions for an elliptic problem with double critical Hardy-Sobolev-Maz'ya terms

 1 School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China, China

Received  November 2014 Revised  April 2015 Published  August 2015

In this paper, by an approximating argument, we obtain infinitely many solutions for the following problem \begin{equation*} \left\{ \begin{array}{ll} -\Delta u = \mu \frac{|u|^{2^{*}(t)-2}u}{|y|^{t}} + \frac{|u|^{2^{*}(s)-2}u}{|y|^{s}} + a(x) u, & \hbox{$\text{in} \Omega$}, \\ u=0,\,\, &\hbox{$\text{on}~\partial \Omega$}, \\ \end{array} \right. \end{equation*} where $\mu\geq0,2^{*}(t)=\frac{2(N-t)}{N-2},2^{*}(s) = \frac{2(N-s)}{N-2},0\leq t < s < 2,x = (y,z)\in \mathbb{R}^{k} \times \mathbb{R}^{N-k},2 \leq k < N,(0,z^*)\subset \bar{\Omega}$ and $\Omega$ is an open bounded domain in $\mathbb{R}^{N}.$ We prove that if $N > 6+t$ when $\mu>0$ and $N > 6+s$ when $\mu=0,$ $a((0,z^*)) > 0,$ $\Omega$ satisfies some geometric conditions, then the above problem has infinitely many solutions.
Citation: Chunhua Wang, Jing Yang. Infinitely many solutions for an elliptic problem with double critical Hardy-Sobolev-Maz'ya terms. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1603-1628. doi: 10.3934/dcds.2016.36.1603
##### References:
 [1] A. Al-aati, C. Wang and J. Zhao, Positive solutions to a semilinear elliptic equation with a Sobolev-Hardy term, Nonlinear Anal., 74 (2011), 4847-4861. doi: 10.1016/j.na.2011.04.057. [2] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Func. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7. [3] A. Bahri and J. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294. doi: 10.1002/cpa.3160410302. [4] M. Bhakta and K. Sandeep, Hardy-Sobolev-Maz'ya type equations in bounded domains, J. Differential Equations, 247 (2009), 119-139. doi: 10.1016/j.jde.2008.12.011. [5] H. Brezis, Nonlinear elliptic equations involving the critical Sobolev exponent-survey and perspectives, in Directions in Partial Differential Equations (Madison, WI, 1985), Publ. Math. Res. Center Univ. Wisconsin, 54, Academic Press, Bonston, MA, 1987, 17-36. [6] H. Brezis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl., 58 (1979), 137-151. [7] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405. [8] D. Cao and P. Han, Solutions to critical elliptic equations with multi-singular inverse square potentials, J. Differential Equations, 224 (2006), 332-372. doi: 10.1016/j.jde.2005.07.010. [9] D. Cao and S. Peng, A global compactness result for singular elliptic problems involving critical Sobolev exponent, Proc. Amer. Math. Soc., 131 (2003), 1857-1866. doi: 10.1090/S0002-9939-02-06729-1. [10] D. Cao and S. Peng, A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms, J. Differential Equations, 193 (2003), 424-434. doi: 10.1016/S0022-0396(03)00118-9. [11] D. Cao, S. Peng and S. Yan, Infinitely many solutions for p-Laplacian equation involving critical Sobolev growth, J. Funct. Anal., 262 (2012), 2861-2902. doi: 10.1016/j.jfa.2012.01.006. [12] D. Cao and S. Yan, Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential, Calc. Var. Partial Differential Equations, 38 (2010), 471-501. doi: 10.1007/s00526-009-0295-5. [13] A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. H. Poinceré Anal. Non Linéaire , 2 (1985), 463-470. [14] D. Castorina, D. Fabbri, G. Mancini and K. Sandeep, Hardy-Sobolev extremals, hyperbolic symmetry and scalar curvature equations, J. Differential Equations, 246 (2009), 1187-1206. doi: 10.1016/j.jde.2008.09.006. [15] G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal., 69 (1986), 289-306. doi: 10.1016/0022-1236(86)90094-7. [16] J. L. Chern and C. S. Lin, Minimizer of Caffarelli-Kohn-Nirenberg inequlities with the singularity on the boundary, Arch. Ration. Mech. Anal., 197 (2010), 401-432. doi: 10.1007/s00205-009-0269-y. [17] G. Devillanova and S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth, Adv. Differential Equations, 7 (2002), 1257-1280. [18] D. Fabbri, G. Mancini and K. Sandeep, Classification of solutions of a critical Hardy-Sobolev operator, J. Differential Equations , 224 (2006), 258-276. doi: 10.1016/j.jde.2005.07.001. [19] N. Ghoussoub and X. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 767-793. doi: 10.1016/j.anihpc.2003.07.002. [20] N. Ghoussoub and F. Robert, The effect of curvature on the best constant in the Hardy-Sobolev inequalities, Geom. Funct. Anal., 16 (2006), 1201-1245. doi: 10.1007/s00039-006-0579-2. [21] N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743. doi: 10.1090/S0002-9947-00-02560-5. [22] C. H. Hsia, C. S. Lin and H. Wadade, Revisiting an idea of Brézis and Nirenberg, J. Funct. Anal., 259 (2010), 1816-1849. doi: 10.1016/j.jfa.2010.05.004. [23] Y. Li and C. S. Lin, A nonlinear elliptic PDE and two Sobolev-Hardy critical exponents, Arch. Ration. Mech. Anal., 203 (2012), 943-968. doi: 10.1007/s00205-011-0467-2. [24] P. L. Lions, The concentration-compactness principle in the calculus of variations, The limit case, part 1 and part 2, Rev. Mat. Iberoamericana, 1 (1985), 145-201; 2 (1985), 45-121. [25] G. Mancini and K. Sandeep, Cylindrical symmetry of extremals of a Hardy-Sobolev inequality, Ann. Mat. Pura Appl., 183 (2004), 165-172. doi: 10.1007/s10231-003-0084-2. [26] G. Mancini and K. Sandeep, On a semilinear elliptic equation in $\mathbbH^n$, Ann. Scuola. Norm. Sup. Pisa Cl. Sci., 7 (2008), 635-671. [27] R. Musina, Ground state soluions of a critical problem involving cylindrical weights, Nonlinear Anal, 68 (2008), 3927-3986. doi: 10.1016/j.na.2007.04.034. [28] S. Peng and C. Wang, Infinitely many solutions for Hardy-Sobolev equation involving critical growth, Math. Meth. Appl. Sci., 38 (2015), 197-220. doi: 10.1002/mma.3060. [29] P. H. Rabinowtz, Minimax Methods in Critical Points Theory with Applications to Differnetial Equations, CBMS Regional Conference Series in Mathematics, 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. [30] M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517. doi: 10.1007/BF01174186. [31] C. Wang and J. Wang, Infinitely many solutions for Hardy-Sobolev-Maz'ya equation involving critical growth, Commun. Contemp. Math., 14 (2012), 1250044, 38pp. doi: 10.1142/S0219199712500447. [32] C. Wang and C. Xiang, Infinitely many solutions for p-Laplacian equation involving double critical terms and boundary geometry, arXiv:1407.7982v2, 2015. [33] S. Yan and J. Yang, Infinitely many solutions for an elliptic problem involving critical Sobolev and Hardy-Sobolev exponents, Calc. Var. Partial Differential Equations, 48 (2013), 587-610. doi: 10.1007/s00526-012-0563-7.

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##### References:
 [1] A. Al-aati, C. Wang and J. Zhao, Positive solutions to a semilinear elliptic equation with a Sobolev-Hardy term, Nonlinear Anal., 74 (2011), 4847-4861. doi: 10.1016/j.na.2011.04.057. [2] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Func. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7. [3] A. Bahri and J. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294. doi: 10.1002/cpa.3160410302. [4] M. Bhakta and K. Sandeep, Hardy-Sobolev-Maz'ya type equations in bounded domains, J. Differential Equations, 247 (2009), 119-139. doi: 10.1016/j.jde.2008.12.011. [5] H. Brezis, Nonlinear elliptic equations involving the critical Sobolev exponent-survey and perspectives, in Directions in Partial Differential Equations (Madison, WI, 1985), Publ. Math. Res. Center Univ. Wisconsin, 54, Academic Press, Bonston, MA, 1987, 17-36. [6] H. Brezis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl., 58 (1979), 137-151. [7] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405. [8] D. Cao and P. Han, Solutions to critical elliptic equations with multi-singular inverse square potentials, J. Differential Equations, 224 (2006), 332-372. doi: 10.1016/j.jde.2005.07.010. [9] D. Cao and S. Peng, A global compactness result for singular elliptic problems involving critical Sobolev exponent, Proc. Amer. Math. Soc., 131 (2003), 1857-1866. doi: 10.1090/S0002-9939-02-06729-1. [10] D. Cao and S. Peng, A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms, J. Differential Equations, 193 (2003), 424-434. doi: 10.1016/S0022-0396(03)00118-9. [11] D. Cao, S. Peng and S. Yan, Infinitely many solutions for p-Laplacian equation involving critical Sobolev growth, J. Funct. Anal., 262 (2012), 2861-2902. doi: 10.1016/j.jfa.2012.01.006. [12] D. Cao and S. Yan, Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential, Calc. Var. Partial Differential Equations, 38 (2010), 471-501. doi: 10.1007/s00526-009-0295-5. [13] A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. H. Poinceré Anal. Non Linéaire , 2 (1985), 463-470. [14] D. Castorina, D. Fabbri, G. Mancini and K. Sandeep, Hardy-Sobolev extremals, hyperbolic symmetry and scalar curvature equations, J. Differential Equations, 246 (2009), 1187-1206. doi: 10.1016/j.jde.2008.09.006. [15] G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal., 69 (1986), 289-306. doi: 10.1016/0022-1236(86)90094-7. [16] J. L. Chern and C. S. Lin, Minimizer of Caffarelli-Kohn-Nirenberg inequlities with the singularity on the boundary, Arch. Ration. Mech. Anal., 197 (2010), 401-432. doi: 10.1007/s00205-009-0269-y. [17] G. Devillanova and S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth, Adv. Differential Equations, 7 (2002), 1257-1280. [18] D. Fabbri, G. Mancini and K. Sandeep, Classification of solutions of a critical Hardy-Sobolev operator, J. Differential Equations , 224 (2006), 258-276. doi: 10.1016/j.jde.2005.07.001. [19] N. Ghoussoub and X. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 767-793. doi: 10.1016/j.anihpc.2003.07.002. [20] N. Ghoussoub and F. Robert, The effect of curvature on the best constant in the Hardy-Sobolev inequalities, Geom. Funct. Anal., 16 (2006), 1201-1245. doi: 10.1007/s00039-006-0579-2. [21] N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743. doi: 10.1090/S0002-9947-00-02560-5. [22] C. H. Hsia, C. S. Lin and H. Wadade, Revisiting an idea of Brézis and Nirenberg, J. Funct. Anal., 259 (2010), 1816-1849. doi: 10.1016/j.jfa.2010.05.004. [23] Y. Li and C. S. Lin, A nonlinear elliptic PDE and two Sobolev-Hardy critical exponents, Arch. Ration. Mech. Anal., 203 (2012), 943-968. doi: 10.1007/s00205-011-0467-2. [24] P. L. Lions, The concentration-compactness principle in the calculus of variations, The limit case, part 1 and part 2, Rev. Mat. Iberoamericana, 1 (1985), 145-201; 2 (1985), 45-121. [25] G. Mancini and K. Sandeep, Cylindrical symmetry of extremals of a Hardy-Sobolev inequality, Ann. Mat. Pura Appl., 183 (2004), 165-172. doi: 10.1007/s10231-003-0084-2. [26] G. Mancini and K. Sandeep, On a semilinear elliptic equation in $\mathbbH^n$, Ann. Scuola. Norm. Sup. Pisa Cl. Sci., 7 (2008), 635-671. [27] R. Musina, Ground state soluions of a critical problem involving cylindrical weights, Nonlinear Anal, 68 (2008), 3927-3986. doi: 10.1016/j.na.2007.04.034. [28] S. Peng and C. Wang, Infinitely many solutions for Hardy-Sobolev equation involving critical growth, Math. Meth. Appl. Sci., 38 (2015), 197-220. doi: 10.1002/mma.3060. [29] P. H. Rabinowtz, Minimax Methods in Critical Points Theory with Applications to Differnetial Equations, CBMS Regional Conference Series in Mathematics, 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. [30] M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517. doi: 10.1007/BF01174186. [31] C. Wang and J. Wang, Infinitely many solutions for Hardy-Sobolev-Maz'ya equation involving critical growth, Commun. Contemp. Math., 14 (2012), 1250044, 38pp. doi: 10.1142/S0219199712500447. [32] C. Wang and C. Xiang, Infinitely many solutions for p-Laplacian equation involving double critical terms and boundary geometry, arXiv:1407.7982v2, 2015. [33] S. Yan and J. Yang, Infinitely many solutions for an elliptic problem involving critical Sobolev and Hardy-Sobolev exponents, Calc. Var. Partial Differential Equations, 48 (2013), 587-610. doi: 10.1007/s00526-012-0563-7.
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